Find the difference between the greatest and least values of the function `f(x)=sin2x-x` on `[-pi/2,pi/2]dot`

Explanation

`f(x)=sin 2 x-x` `f^{prime}(x)=2 cos 2 x-1=0`

`Rightarrow cos 2 x=frac{1}{2}`

`Rightarrow cos 2 x=cos frac{pi}{3}`

`x=frac{pi}{6}`

`f^{prime prime}(x)=-2 sin 2 x<0` put `x=frac{pi}{6}`

`f^{prime prime}(x)=-2 sin frac{pi}{2}=2 times frac{sqrt{3}}{2}=-sqrt{3}<0`

`Rightarrow x=frac{pi}{6}` point pof maxima

maximum value

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